Stochastic models of infection dynamics
Ana Nunes
Centro de Física Teórica e Computacional

( Here you can find the slides of the presentation)

Abstract

During the last decade, more sophisticated approaches building on the traditional SIR and SEIR models have brought considerable advances in understanding and selecting some of the fundamental ingredients of the complex dynamics of infectious diseases [1]. This body of work belongs to an essentially deterministic framework, where demographic stochasticity plays a secondary role, that of sustaining small amplitude fluctuations around the deterministic system’s equilibrium that follow the natural frequency given by the local linear approximation [2].

In population biology, stochasticity comes from the discrete interactions and the disordered interaction networks, and fluctuations and finite size effects in general are much more important than in typical physical systems. Indeed, recent results show that for realistic population sizes, the behaviour of predator-prey and of epidemic or endemic infections may be driven by the combined effect of fluctuations and correlations.

In [3], a general mechanism of resonant amplification of demographic stochasticity was proposed to describe the cycling behaviour of prey-predator systems. This resonant mechanism is generic for a class of stochastic systems that includes the majority of the classical models of diseases that confer either lifelong or temporary total immunity, and it was shown in [4] that it plays a major role in describing the patterns of recurrent epidemics of childhood infectious diseases. In [5], that approach was exended by adding an ingredient which is missing in standard epidemic models, the 'mixing network' through which infection may propagate. It was shown that correlations have a major effect in the enhancement of the amplitude and the coherence of the resonant stochastic fluctuations, providing ordered patterns of recurrent epidemics, whose period may differ significantly from that of the small oscillations around the deterministic equilibrium.

We shall review the main results of [1-5], and then explore analytic models where the assumptions of random mixing of the population and/or of constant recovery rate during the infectious period are relaxed, and see how this implies important corrections to the amplitude and dominant frequency of the stochastic fluctuations [6-8]. The finding that in finite, discrete populations internal noise together with correlations produces sustained incidence oscillations of significant amplitude all over the parameter region that includes childhood infectious diseases is of importance for the long-standing controversy in epidemiology and ecology as to the driving mechanisms of the pervasive noisy oscillations observed in these systems.

[1] D. J. D. Earn, P. Rohani, B. M. Bolker and B. T. Grenfell, A Simple Model for Complex Dynamical Transitions in Epidemics, Science 287, 667-670 (2000);
[2] C. T. Bausch and D. J. D. Earn, Transients and attractors in epidemics, Proc. R. Soc. Lond. B 270, 1573-1578 (2003);
[3] A.J. McKane, T.J. Newman, "Predator-Prey Cycles from Resonant Amplification of Demographic Stochasticity", Physical Review Letters, 94, 218102 (2005);
[4] D. Alonso, A. J. McKane, M. Pascual, "Stochastic amplification in epidemics", J R Soc Interface 4, 575-82 (2007);
[5] M. Simões, M. Telo da Gama and A. Nunes, "Stochastic fluctuations in epidemics on networks", J R Soc Interface 5, 555-66 (2008);
[6] Ganna Rozhnova & Ana Nunes, “Fluctuations and oscillations in a simple epidemic model”, Phys. Rev. E 79, 041922 (2009) & Virtual Journal of Biological Physics Research 17 (9), May-1 (2009);
[7] Andrew J. Black, Alan J. McKane, Ana Nunes & Andrea Parisi , “Stochastic fluctuations in the susceptible-infective-recovered model with distributed infectious periods”, Phys. Rev. E 80, 021922 (2009);
[8] Ganna Rozhnova & Ana Nunes, “Cluster approximations for infection dynamics on random networks”, Phys. Rev. E 80, 051915 (2009).

Jornada Matemática SPM/CIM em Epidemiologia Teórica